If sin 40° = p, write EACH of the following in terms of p. 5.1.1 sin 220° 5.1.2 cos² 50° 5.1.3 cos(−80°) Given: tan x(1−cos² x) + cos² x = \( \frac{(sin x + cos x)(1−sin x cos x)}{cos x} \) 5.2.1 Prove the above identity. 5.2.2 For which values of x, in the interval \( x ∈ [-180°; 180°] \), will the identity be undefined? 5.3 Given the expression: \( \frac{sin 150° + cos² x - 1}{2} \) 5.3.1 Without using a calculator, simplify the expression given above to a single trigonometric term in terms of cos 2x. 5.3.2 Hence, determine the general solution of \( \frac{sin 150° + cos² x - 1}{25} \)

NSC Mathematics Trigonometry: If sin 40° = p, write EACH of th

If sin 40° = p, write EACH of the following in terms of p - NSC Mathematics - Question 5 - 2024 - Paper 2

Question 5

If sin 40° = p, write EACH of the following in terms of p. 5.1.1 sin 220° 5.1.2 cos² 50° 5.1.3 cos(−80°) Given: tan x(1−cos² x) + cos² x = \( \frac{(sin x + cos x... show full transcript

Worked Solution & Example Answer:If sin 40° = p, write EACH of the following in terms of p - NSC Mathematics - Question 5 - 2024 - Paper 2

Step 1

5.1.1 sin 220°

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Answer

Using the sine identity, we can rewrite sin 220° as:

sin 220° = sin(180° + 40°) = -sin 40° = -p

Step 2

5.1.2 cos² 50°

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Answer

For cos² 50°, we can express it as:

cos² 50° = 1 - sin² 50°

Knowing that sin 50° can be obtained through the angle subtraction identity:

sin 50° = sin(90° - 40°) = cos 40° = \sqrt{1 - p^2}

Thus,

cos² 50° = 1 - (1 - p^2) = p^2

Step 3

5.1.3 cos(−80°)

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Answer

For cos(−80°), we use the even property of cosine:

cos(−80°) = cos(80°)

Using the sine identity again, we find:

cos(80° = cos(90° - 10°) = sin 10°

Due to known angles, we express this in terms of p:

cos(80°) = √(1 - sin² 10°)

Step 4

5.2.1 Prove the above identity.

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Answer

To prove the identity, start with the left-hand side (LHS):

tan x(1 - cos² x) + cos² x = \frac{sin x}{cos x} (1 - cos² x) + cos² x

Substituting the identities:

= \frac{sin x(1 - cos² x + cos² x)}{cos x} = \frac{sin x}{cos x}

Thus, LHS = RHS.

Step 5

5.2.2 For which values of x will the identity be undefined?

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Answer

The identity will be undefined where cos x = 0, which gives:

x = 90° + k * 180°, \text{ for } k ∈ Z

Step 6

5.3.1 Without using a calculator, simplify the expression given above to a single trigonometric term in terms of cos 2x.

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Answer

Start with:

Using the double angle formula for cosine:

We can substitute:

Step 7

5.3.2 Hence, determine the general solution of \( \frac{sin 150° + cos² x - 1}{25} \)

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Answer

Using the result from 5.3.1:

Multiplying both sides by 25:

If sin 40° = p, write EACH of the following in terms of p - NSC Mathematics - Question 5 - 2024 - Paper 2

Worked Solution & Example Answer:If sin 40° = p, write EACH of the following in terms of p - NSC Mathematics - Question 5 - 2024 - Paper 2

5.1.1 sin 220°

5.1.2 cos² 50°

5.1.3 cos(−80°)

5.2.1 Prove the above identity.

5.2.2 For which values of x will the identity be undefined?

5.3.1 Without using a calculator, simplify the expression given above to a single trigonometric term in terms of cos 2x.

5.3.2 Hence, determine the general solution of \( \frac{sin 150° + cos² x - 1}{25} \)

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